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Group Representation


A representation of a group G is a group action of G on a vector space V by invertible linear maps. For example, the group of two elements Z_2={0,1} has a representation phi by phi(0)v=v and phi(1)v=-v. A representation is a group homomorphism phi:G->GL(V).

Most groups have many different representations, possibly on different vector spaces. For example, the symmetric group S_3={e,(12),(13),(23),(123),(132)} has a representation on R by

 phi_1(sigma)v=sgn(sigma)v,
(1)

where sgn(sigma) is the permutation symbol of the permutation sigma. It also has a representation on R^3 by

 phi_2(sigma)(x_1,x_2,x_3)=(x_(sigma(1)),x_(sigma(2)),x_(sigma(3))).
(2)

A representation gives a matrix for each element, and so another representation of S_3 is given by the matrices

 [1 0; 0 1],[0 1; 1 0],[-1 0; -1 1],[1 -1; 0 -1],[-1 1; -1 0],[0 -1; 1 -1].
(3)

Two representations are considered equivalent if they are similar. For example, performing similarity transformations of the above matrices by

 [1 19; 0 1]
(4)

gives the following equivalent representation of S_3,

 [1 0; 0 1],[-19 -360; 1 19],[18 323; -1 -18],[1 37; 0 -1],[18 343; -1 -19],[-19 -343; 1 18].
(5)

Any representation V of G can be restricted to a representation of any subgroup H, in which case, it is denoted Res_H^G. More surprisingly, any representation W on H can be extended to a representation of G, on a larger vector space V, called the induced representation.

Representations have applications to many branches of mathematics, aside from applications to physics and chemistry. The name of the theory depends on the group G and on the vector space V. Different approaches are required depending on whether G is a finite group, an infinite discrete group, or a Lie group. Another important ingredient is the field of scalars for V. The vector space V can be infinite dimensional such as a Hilbert space. Also, special kinds of representations may require that a vector space structure is preserved. For instance, a unitary representation is a group homomorphism phi:G->U(V) into the group of unitary transformations which preserve a Hermitian inner product on V.

In favorable situations, such as a finite group, an arbitrary representation will break up into irreducible representations, i.e., V= direct sum V_i where the V_i are irreducible. For many groups, the irreducible representations have been classified.


See also

Group, Irreducible Representation, Lie Algebra Representation, Multiplicative Character, Orthogonal Group Representations, Peter-Weyl Theorem, Primary Representation, Representation Ring, Representation Tensor Product, Representation Theory, Schur's Lemma, Semisimple Lie Group, Unitary Representation, Vector Space Explore this topic in the MathWorld classroom

This entry contributed by Todd Rowland

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Rowland, Todd. "Group Representation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/GroupRepresentation.html

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